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Theorem scshwfzeqfzo 30492
Description: Lemma1: For a nonempty word the sets of shifted words, expressd by a finite interval of integers or by a half-open integer range are identical. (Contributed by Alexander van der Vekens, 15-Jun-2018.)
Assertion
Ref Expression
scshwfzeqfzo  |-  ( ( X  e. Word  V  /\  X  =/=  (/)  /\  N  =  ( # `  X
) )  ->  { y  e. Word  V  |  E. n  e.  ( 0 ... N ) y  =  ( X cyclShift  n ) }  =  { y  e. Word  V  |  E. n  e.  ( 0..^ N ) y  =  ( X cyclShift  n ) } )
Distinct variable groups:    n, N, y    n, V, y    n, X, y

Proof of Theorem scshwfzeqfzo
StepHypRef Expression
1 lencl 12249 . . . . . . . . . . . 12  |-  ( X  e. Word  V  ->  ( # `
 X )  e. 
NN0 )
2 elnn0uz 10898 . . . . . . . . . . . 12  |-  ( (
# `  X )  e.  NN0  <->  ( # `  X
)  e.  ( ZZ>= ` 
0 ) )
31, 2sylib 196 . . . . . . . . . . 11  |-  ( X  e. Word  V  ->  ( # `
 X )  e.  ( ZZ>= `  0 )
)
43adantr 465 . . . . . . . . . 10  |-  ( ( X  e. Word  V  /\  N  =  ( # `  X
) )  ->  ( # `
 X )  e.  ( ZZ>= `  0 )
)
5 eleq1 2503 . . . . . . . . . . 11  |-  ( N  =  ( # `  X
)  ->  ( N  e.  ( ZZ>= `  0 )  <->  (
# `  X )  e.  ( ZZ>= `  0 )
) )
65adantl 466 . . . . . . . . . 10  |-  ( ( X  e. Word  V  /\  N  =  ( # `  X
) )  ->  ( N  e.  ( ZZ>= ` 
0 )  <->  ( # `  X
)  e.  ( ZZ>= ` 
0 ) ) )
74, 6mpbird 232 . . . . . . . . 9  |-  ( ( X  e. Word  V  /\  N  =  ( # `  X
) )  ->  N  e.  ( ZZ>= `  0 )
)
873adant2 1007 . . . . . . . 8  |-  ( ( X  e. Word  V  /\  X  =/=  (/)  /\  N  =  ( # `  X
) )  ->  N  e.  ( ZZ>= `  0 )
)
98adantr 465 . . . . . . 7  |-  ( ( ( X  e. Word  V  /\  X  =/=  (/)  /\  N  =  ( # `  X
) )  /\  y  e. Word  V )  ->  N  e.  ( ZZ>= `  0 )
)
10 fzisfzounsn 30211 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  0
)  ->  ( 0 ... N )  =  ( ( 0..^ N )  u.  { N } ) )
119, 10syl 16 . . . . . 6  |-  ( ( ( X  e. Word  V  /\  X  =/=  (/)  /\  N  =  ( # `  X
) )  /\  y  e. Word  V )  ->  (
0 ... N )  =  ( ( 0..^ N )  u.  { N } ) )
1211rexeqdv 2924 . . . . 5  |-  ( ( ( X  e. Word  V  /\  X  =/=  (/)  /\  N  =  ( # `  X
) )  /\  y  e. Word  V )  ->  ( E. n  e.  (
0 ... N ) y  =  ( X cyclShift  n )  <->  E. n  e.  (
( 0..^ N )  u.  { N }
) y  =  ( X cyclShift  n ) ) )
13 rexun 3536 . . . . 5  |-  ( E. n  e.  ( ( 0..^ N )  u. 
{ N } ) y  =  ( X cyclShift  n )  <->  ( E. n  e.  ( 0..^ N ) y  =  ( X cyclShift  n )  \/  E. n  e.  { N } y  =  ( X cyclShift  n ) ) )
1412, 13syl6bb 261 . . . 4  |-  ( ( ( X  e. Word  V  /\  X  =/=  (/)  /\  N  =  ( # `  X
) )  /\  y  e. Word  V )  ->  ( E. n  e.  (
0 ... N ) y  =  ( X cyclShift  n )  <-> 
( E. n  e.  ( 0..^ N ) y  =  ( X cyclShift  n )  \/  E. n  e.  { N } y  =  ( X cyclShift  n ) ) ) )
15 ax-1 6 . . . . . 6  |-  ( E. n  e.  ( 0..^ N ) y  =  ( X cyclShift  n )  ->  ( ( ( X  e. Word  V  /\  X  =/=  (/)  /\  N  =  ( # `  X
) )  /\  y  e. Word  V )  ->  E. n  e.  ( 0..^ N ) y  =  ( X cyclShift  n ) ) )
16 fvex 5701 . . . . . . . . . . . . 13  |-  ( # `  X )  e.  _V
1716a1i 11 . . . . . . . . . . . 12  |-  ( N  =  ( # `  X
)  ->  ( # `  X
)  e.  _V )
18 eleq1 2503 . . . . . . . . . . . 12  |-  ( N  =  ( # `  X
)  ->  ( N  e.  _V  <->  ( # `  X
)  e.  _V )
)
1917, 18mpbird 232 . . . . . . . . . . 11  |-  ( N  =  ( # `  X
)  ->  N  e.  _V )
20 oveq2 6099 . . . . . . . . . . . . 13  |-  ( n  =  N  ->  ( X cyclShift  n )  =  ( X cyclShift  N ) )
2120eqeq2d 2454 . . . . . . . . . . . 12  |-  ( n  =  N  ->  (
y  =  ( X cyclShift  n )  <->  y  =  ( X cyclShift  N ) ) )
2221rexsng 3913 . . . . . . . . . . 11  |-  ( N  e.  _V  ->  ( E. n  e.  { N } y  =  ( X cyclShift  n )  <->  y  =  ( X cyclShift  N ) ) )
2319, 22syl 16 . . . . . . . . . 10  |-  ( N  =  ( # `  X
)  ->  ( E. n  e.  { N } y  =  ( X cyclShift  n )  <->  y  =  ( X cyclShift  N ) ) )
24233ad2ant3 1011 . . . . . . . . 9  |-  ( ( X  e. Word  V  /\  X  =/=  (/)  /\  N  =  ( # `  X
) )  ->  ( E. n  e.  { N } y  =  ( X cyclShift  n )  <->  y  =  ( X cyclShift  N ) ) )
2524adantr 465 . . . . . . . 8  |-  ( ( ( X  e. Word  V  /\  X  =/=  (/)  /\  N  =  ( # `  X
) )  /\  y  e. Word  V )  ->  ( E. n  e.  { N } y  =  ( X cyclShift  n )  <->  y  =  ( X cyclShift  N ) ) )
26 oveq2 6099 . . . . . . . . . . . . 13  |-  ( N  =  ( # `  X
)  ->  ( X cyclShift  N )  =  ( X cyclShift  ( # `  X ) ) )
27263ad2ant3 1011 . . . . . . . . . . . 12  |-  ( ( X  e. Word  V  /\  X  =/=  (/)  /\  N  =  ( # `  X
) )  ->  ( X cyclShift  N )  =  ( X cyclShift  ( # `  X
) ) )
28 cshwn 12434 . . . . . . . . . . . . 13  |-  ( X  e. Word  V  ->  ( X cyclShift  ( # `  X
) )  =  X )
29283ad2ant1 1009 . . . . . . . . . . . 12  |-  ( ( X  e. Word  V  /\  X  =/=  (/)  /\  N  =  ( # `  X
) )  ->  ( X cyclShift  ( # `  X
) )  =  X )
3027, 29eqtrd 2475 . . . . . . . . . . 11  |-  ( ( X  e. Word  V  /\  X  =/=  (/)  /\  N  =  ( # `  X
) )  ->  ( X cyclShift  N )  =  X )
3130eqeq2d 2454 . . . . . . . . . 10  |-  ( ( X  e. Word  V  /\  X  =/=  (/)  /\  N  =  ( # `  X
) )  ->  (
y  =  ( X cyclShift  N )  <->  y  =  X ) )
3231adantr 465 . . . . . . . . 9  |-  ( ( ( X  e. Word  V  /\  X  =/=  (/)  /\  N  =  ( # `  X
) )  /\  y  e. Word  V )  ->  (
y  =  ( X cyclShift  N )  <->  y  =  X ) )
33 cshw0 12431 . . . . . . . . . . . . . . 15  |-  ( X  e. Word  V  ->  ( X cyclShift  0 )  =  X )
34333ad2ant1 1009 . . . . . . . . . . . . . 14  |-  ( ( X  e. Word  V  /\  X  =/=  (/)  /\  N  =  ( # `  X
) )  ->  ( X cyclShift  0 )  =  X )
35 lennncl 12250 . . . . . . . . . . . . . . . . . 18  |-  ( ( X  e. Word  V  /\  X  =/=  (/) )  ->  ( # `
 X )  e.  NN )
36353adant3 1008 . . . . . . . . . . . . . . . . 17  |-  ( ( X  e. Word  V  /\  X  =/=  (/)  /\  N  =  ( # `  X
) )  ->  ( # `
 X )  e.  NN )
37 eleq1 2503 . . . . . . . . . . . . . . . . . 18  |-  ( N  =  ( # `  X
)  ->  ( N  e.  NN  <->  ( # `  X
)  e.  NN ) )
38373ad2ant3 1011 . . . . . . . . . . . . . . . . 17  |-  ( ( X  e. Word  V  /\  X  =/=  (/)  /\  N  =  ( # `  X
) )  ->  ( N  e.  NN  <->  ( # `  X
)  e.  NN ) )
3936, 38mpbird 232 . . . . . . . . . . . . . . . 16  |-  ( ( X  e. Word  V  /\  X  =/=  (/)  /\  N  =  ( # `  X
) )  ->  N  e.  NN )
40 lbfzo0 11586 . . . . . . . . . . . . . . . 16  |-  ( 0  e.  ( 0..^ N )  <->  N  e.  NN )
4139, 40sylibr 212 . . . . . . . . . . . . . . 15  |-  ( ( X  e. Word  V  /\  X  =/=  (/)  /\  N  =  ( # `  X
) )  ->  0  e.  ( 0..^ N ) )
42 oveq2 6099 . . . . . . . . . . . . . . . . . . . 20  |-  ( 0  =  n  ->  ( X cyclShift  0 )  =  ( X cyclShift  n ) )
4342eqeq1d 2451 . . . . . . . . . . . . . . . . . . 19  |-  ( 0  =  n  ->  (
( X cyclShift  0 )  =  X  <->  ( X cyclShift  n )  =  X ) )
4443eqcoms 2446 . . . . . . . . . . . . . . . . . 18  |-  ( n  =  0  ->  (
( X cyclShift  0 )  =  X  <->  ( X cyclShift  n )  =  X ) )
45 eqcom 2445 . . . . . . . . . . . . . . . . . 18  |-  ( ( X cyclShift  n )  =  X  <-> 
X  =  ( X cyclShift  n ) )
4644, 45syl6bb 261 . . . . . . . . . . . . . . . . 17  |-  ( n  =  0  ->  (
( X cyclShift  0 )  =  X  <->  X  =  ( X cyclShift  n ) ) )
4746adantl 466 . . . . . . . . . . . . . . . 16  |-  ( ( ( X  e. Word  V  /\  X  =/=  (/)  /\  N  =  ( # `  X
) )  /\  n  =  0 )  -> 
( ( X cyclShift  0
)  =  X  <->  X  =  ( X cyclShift  n ) ) )
4847biimpd 207 . . . . . . . . . . . . . . 15  |-  ( ( ( X  e. Word  V  /\  X  =/=  (/)  /\  N  =  ( # `  X
) )  /\  n  =  0 )  -> 
( ( X cyclShift  0
)  =  X  ->  X  =  ( X cyclShift  n ) ) )
4941, 48rspcimedv 3075 . . . . . . . . . . . . . 14  |-  ( ( X  e. Word  V  /\  X  =/=  (/)  /\  N  =  ( # `  X
) )  ->  (
( X cyclShift  0 )  =  X  ->  E. n  e.  ( 0..^ N ) X  =  ( X cyclShift  n ) ) )
5034, 49mpd 15 . . . . . . . . . . . . 13  |-  ( ( X  e. Word  V  /\  X  =/=  (/)  /\  N  =  ( # `  X
) )  ->  E. n  e.  ( 0..^ N ) X  =  ( X cyclShift  n ) )
5150adantr 465 . . . . . . . . . . . 12  |-  ( ( ( X  e. Word  V  /\  X  =/=  (/)  /\  N  =  ( # `  X
) )  /\  y  e. Word  V )  ->  E. n  e.  ( 0..^ N ) X  =  ( X cyclShift  n ) )
5251adantr 465 . . . . . . . . . . 11  |-  ( ( ( ( X  e. Word  V  /\  X  =/=  (/)  /\  N  =  ( # `  X
) )  /\  y  e. Word  V )  /\  y  =  X )  ->  E. n  e.  ( 0..^ N ) X  =  ( X cyclShift  n ) )
53 eqeq1 2449 . . . . . . . . . . . . 13  |-  ( y  =  X  ->  (
y  =  ( X cyclShift  n )  <->  X  =  ( X cyclShift  n ) ) )
5453adantl 466 . . . . . . . . . . . 12  |-  ( ( ( ( X  e. Word  V  /\  X  =/=  (/)  /\  N  =  ( # `  X
) )  /\  y  e. Word  V )  /\  y  =  X )  ->  (
y  =  ( X cyclShift  n )  <->  X  =  ( X cyclShift  n ) ) )
5554rexbidv 2736 . . . . . . . . . . 11  |-  ( ( ( ( X  e. Word  V  /\  X  =/=  (/)  /\  N  =  ( # `  X
) )  /\  y  e. Word  V )  /\  y  =  X )  ->  ( E. n  e.  (
0..^ N ) y  =  ( X cyclShift  n )  <->  E. n  e.  (
0..^ N ) X  =  ( X cyclShift  n ) ) )
5652, 55mpbird 232 . . . . . . . . . 10  |-  ( ( ( ( X  e. Word  V  /\  X  =/=  (/)  /\  N  =  ( # `  X
) )  /\  y  e. Word  V )  /\  y  =  X )  ->  E. n  e.  ( 0..^ N ) y  =  ( X cyclShift  n ) )
5756ex 434 . . . . . . . . 9  |-  ( ( ( X  e. Word  V  /\  X  =/=  (/)  /\  N  =  ( # `  X
) )  /\  y  e. Word  V )  ->  (
y  =  X  ->  E. n  e.  (
0..^ N ) y  =  ( X cyclShift  n ) ) )
5832, 57sylbid 215 . . . . . . . 8  |-  ( ( ( X  e. Word  V  /\  X  =/=  (/)  /\  N  =  ( # `  X
) )  /\  y  e. Word  V )  ->  (
y  =  ( X cyclShift  N )  ->  E. n  e.  ( 0..^ N ) y  =  ( X cyclShift  n ) ) )
5925, 58sylbid 215 . . . . . . 7  |-  ( ( ( X  e. Word  V  /\  X  =/=  (/)  /\  N  =  ( # `  X
) )  /\  y  e. Word  V )  ->  ( E. n  e.  { N } y  =  ( X cyclShift  n )  ->  E. n  e.  ( 0..^ N ) y  =  ( X cyclShift  n ) ) )
6059com12 31 . . . . . 6  |-  ( E. n  e.  { N } y  =  ( X cyclShift  n )  ->  (
( ( X  e. Word  V  /\  X  =/=  (/)  /\  N  =  ( # `  X
) )  /\  y  e. Word  V )  ->  E. n  e.  ( 0..^ N ) y  =  ( X cyclShift  n ) ) )
6115, 60jaoi 379 . . . . 5  |-  ( ( E. n  e.  ( 0..^ N ) y  =  ( X cyclShift  n )  \/  E. n  e. 
{ N } y  =  ( X cyclShift  n ) )  ->  ( (
( X  e. Word  V  /\  X  =/=  (/)  /\  N  =  ( # `  X
) )  /\  y  e. Word  V )  ->  E. n  e.  ( 0..^ N ) y  =  ( X cyclShift  n ) ) )
6261com12 31 . . . 4  |-  ( ( ( X  e. Word  V  /\  X  =/=  (/)  /\  N  =  ( # `  X
) )  /\  y  e. Word  V )  ->  (
( E. n  e.  ( 0..^ N ) y  =  ( X cyclShift  n )  \/  E. n  e.  { N } y  =  ( X cyclShift  n ) )  ->  E. n  e.  (
0..^ N ) y  =  ( X cyclShift  n ) ) )
6314, 62sylbid 215 . . 3  |-  ( ( ( X  e. Word  V  /\  X  =/=  (/)  /\  N  =  ( # `  X
) )  /\  y  e. Word  V )  ->  ( E. n  e.  (
0 ... N ) y  =  ( X cyclShift  n )  ->  E. n  e.  ( 0..^ N ) y  =  ( X cyclShift  n ) ) )
64 fzossfz 11570 . . . 4  |-  ( 0..^ N )  C_  (
0 ... N )
65 ssrexv 3417 . . . 4  |-  ( ( 0..^ N )  C_  ( 0 ... N
)  ->  ( E. n  e.  ( 0..^ N ) y  =  ( X cyclShift  n )  ->  E. n  e.  ( 0 ... N ) y  =  ( X cyclShift  n ) ) )
6664, 65mp1i 12 . . 3  |-  ( ( ( X  e. Word  V  /\  X  =/=  (/)  /\  N  =  ( # `  X
) )  /\  y  e. Word  V )  ->  ( E. n  e.  (
0..^ N ) y  =  ( X cyclShift  n )  ->  E. n  e.  ( 0 ... N ) y  =  ( X cyclShift  n ) ) )
6763, 66impbid 191 . 2  |-  ( ( ( X  e. Word  V  /\  X  =/=  (/)  /\  N  =  ( # `  X
) )  /\  y  e. Word  V )  ->  ( E. n  e.  (
0 ... N ) y  =  ( X cyclShift  n )  <->  E. n  e.  (
0..^ N ) y  =  ( X cyclShift  n ) ) )
6867rabbidva 2963 1  |-  ( ( X  e. Word  V  /\  X  =/=  (/)  /\  N  =  ( # `  X
) )  ->  { y  e. Word  V  |  E. n  e.  ( 0 ... N ) y  =  ( X cyclShift  n ) }  =  { y  e. Word  V  |  E. n  e.  ( 0..^ N ) y  =  ( X cyclShift  n ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2606   E.wrex 2716   {crab 2719   _Vcvv 2972    u. cun 3326    C_ wss 3328   (/)c0 3637   {csn 3877   ` cfv 5418  (class class class)co 6091   0cc0 9282   NNcn 10322   NN0cn0 10579   ZZ>=cuz 10861   ...cfz 11437  ..^cfzo 11548   #chash 12103  Word cword 12221   cyclShift ccsh 12425
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-sup 7691  df-card 8109  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-n0 10580  df-z 10647  df-uz 10862  df-rp 10992  df-fz 11438  df-fzo 11549  df-fl 11642  df-mod 11709  df-hash 12104  df-word 12229  df-concat 12231  df-substr 12233  df-csh 12426
This theorem is referenced by:  hashecclwwlkn1  30508  usghashecclwwlk  30509
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